[Jump to code hierarchy]

Bosonic code

Alternative names: Continuous-variable (CV) quantum code, Oscillator code, Quantum modulation scheme.
Root code for the Bosonic Kingdom

Description

Encodes logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space that contains at least one oscillator (a.k.a. bosonic mode or qumode).

States of a single oscillator correspond to \(L^2\)-normalizable functions on \(\mathbb{R}\) that have finite energy, finite variance, and finite values of all other moments (where the energy operator is defined to be the harmonic oscillator Hamiltonian); such functions form Schwartz space (a.k.a. nuclear space [1]), a subspace of Hilbert space [2]. Ideal codewords may not be normalizable because the space is infinite-dimensional, so approximate versions have to be constructed in practice.

States can be represented by a series via a basis expansion, such as that in the countable basis of Fock states \(|n\rangle\) with \(n\geq 0\). Alternatively, states can be represented as functions over the reals by expanding in a continuous "basis" (more technically, set of tempered distributions in the space dual to Schwartz space), such as the position "basis" \(|y\rangle\) with \(y\in\mathbb{R}\) or the momentum "basis" \(|p\rangle\) with \(p\in\mathbb{R}\). A third option is to use coherent states \(|\alpha\rangle\) with \(\alpha\in\mathbb{C}\), which are eigenstates of the annihilation operator, which correspond to classical electromagnetic signals, and which resolve the identity [3–6]. States can further be represented as functions over the joint position-momentum phase space in the Wigner function formalism [7,8]. GKP states have negative Wigner functions, but the alternative Zak-Gross Wigner function represents them positively [9].

An important subset of states is formed by the Gaussian states, which are in one-to-one correspondence with a (displacement) vector and covariance matrix [10–17]. Pure Gaussian states can be obtained from the vacuum Fock state \(|n=0\rangle\) via a Gaussian unitary transformation (defined below). Any coherent state can be obtained from the vacuum Fock state, itself a coherent state, by a displacement.

Protection

Displacement error basis

An error set relevant to bosonic stabilizer codes is the set of displacement operators, a bosonic analogue of the Pauli string basis for qubit codes.

Displacement operators: For a single mode, its elements are products of exponentials of the mode's position and momentum operators, acting on the mode's position states \(|y\rangle\) for \(y\in\mathbb{R}\) as \begin{align} e^{-iq\hat{p}}\left|y\right\rangle =\left|y+q\right\rangle \,\,\text{ and }\,\,e^{iq\hat{x}}\left|y\right\rangle =e^{iq y}\left|y\right\rangle ~, \tag*{(1)}\end{align} where \(q\in\mathbb{R}\). The former is also called a translation, while the latter is called a modulation in signal processing. For multiple modes, error set elements are tensor products of elements of the single-oscillator error set, characterized by the vector of coefficients \(\xi\in\mathbb{R}^{2n}\).

The displacement error set is a unitary basis for bounded operators on the \(n\)-mode Hilbert space that is Dirac-orthonormal under the Hilbert-Schmidt inner product. Expanding a bounded operator in terms of displacements is called the Fourier-Weyl transform (a.k.a. Fourier-Weyl relation) [19][18; Eq. (4.11)]. For the expansion of Gaussian unitary operations in terms of displacements, see [20; Eq. (7.62)].

There are two definitions of code distance associated with displacements. The definition inherited from qubit codes is the minimum weight of a displacement operator (i.e., number of nonzero entries in \(\xi\)) that implements a nontrivial logical operation in the code. The second definition is the minimum Euclidean distance (i.e., \(\ell^2\)-norm of \(\xi\)) such that the corresponding displacement implements a nontrivial logical operation in the code. Quantum weight enumerators and the Cohn-Elkies bound have been extended to the case of displacement noise [21].

Loss and gain operators

An error set relevant to Fock-state bosonic codes is the set of loss operators associated with the AD channel, a common form of physical noise in bosonic systems. For a single mode, loss operators are proportional to powers of the mode's annihilation operator \(a=(\hat{x}+i\hat{p})/\sqrt{2}\), where \(\hat x\) (\(\hat p\)) is the mode's position (momentum) operator, and with the power signifying the number of particles lost during the error. For multiple modes, error set elements are tensor products of elements of the single-mode error set. Quantum Hamming bounds have been extended to the case of loss noise [22]

Number-phase operators

An related error set is the set of powers of the Susskind–Glogower phase operator \(\frac{1}{\sqrt{a a^\dagger}} a\) and its adjoint [23–25] along with Fock-space rotations generated by the occupation number operator \(a^\dagger a\). These can also be obtained from qudit Pauli matrices through a limiting procedure [25] and allow one to expand trace-class operators despite not forming an orthonormal set [2]. These operators are correspond to the number-phase interpretation, a polar-like decomposition of a single mode, complementing the cartesian-like decomposition in terms of position and momentum displacements. This decomposition can be called a number-phase rotor, which differs from the ordinary \(U(1)\) rotor in the absence of states of negative angular momentum. Mathematically, the restriction of an ordinary rotor to states of non-negative momentum is a projection onto Hardy space, and the phase operator is an example of a Toeplitz operator on that space [26].

Rate

The quantum capacity of the AD channel [27] and the dephasing noise channel [28] are both known. The capacity of the displacement noise channel, the quantum analogue of AGWN, has been bounded using GKP codes [29,30]. Exact two-way assisted capacities have been obtained for the AD channels and quantum limited amplifiers in what is known as the PLOB bound [31]. These are examples of Gaussian channels, i.e., channels that map Gaussian states to Gaussian states [32–39]. Bounds exist on the two-way quantum and secret-key capacities for some prominent Gaussian channels [40–49] and non-Markovian channels, i.e., channels with memory effects [50]. Non-Gaussian channel capacities can be bounded for single [51] and multiple [52; Lemma 14] modes. Non-asymptotic bounds exist for memoryless channels [53] as well as for those with memory effects [54].

Gates

Displacement operations form a group called the Heisenberg-Weyl group, the oscillator analogue to the Pauli group. Analogues of (non-Pauli) Clifford-group transformations are the Gaussian unitary transformations (a.k.a. symplectic, Bogoliubov-Valatin, or linear canonical transformations) [15,55,56], which are unitaries generated by quadratic polynomials in positions and momenta. The Gaussian unitary transformation group permutes displacement operators amongst themselves, and, up to any phases, is equivalent to the symplectic group \(Sp(2n,\mathbb{R})\).Computing using Gaussian states and Gaussian unitaries only can be efficiently simulated on a classical computer [57–59], and there are efficient algorithms to do so [12,60]. This remains true even if superpositions of Gaussian states are considered [61,62], but is no longer the case when the number of modes scales exponentially [63].A gate generated by a cubic of higher-degree polynomial is required to make a universal gate set on the oscillator (an infinite-dimensional version of the Solovay-Kitaev theorem) [11,64,65]. The cubic phase gate [66] is a common gate used in tandem with Gaussian gates for universality, and Fock-state matrix elements of the cubic and quartic phase gates have been derived [67] (see also Refs. [68,69]). Arbitrary-degree polynomial gates are well defined, but cubic or higher versions of squeezing admit subtle mathematical properties [70,71]. Unitaries generated by polynomials of position and momentum can exactly realize any finite-dimensional unitary evolution, and any physical bosonic unitary evolution can be approximated by a finite-dimensional unitary evolution [65,72]. See Ref. [73] for bosonic computational complexity classes. The stellar rank quantifies the degree of non-Gaussianity of single-mode states [74]. Functional MPS can be used to simulate evolution of non-Gaussian states [75]. Controllability of bosonic states has been proven when the normalizable state space is restricted to Schwartz space [76] and using polynomials in position and momentum [65].Measurements can be performed by homodyne, heterodyne, and generalized homodyne measurements [77].The number-phase interpretation allows for the mapping of rotor Clifford gates into the oscillator, some of which become non-unitary (e.g., conditional occupation number addition) [78].ZX calculus has been extended to bosonic codes for both Gaussian operators [79] and Fock-state based operators [80]. An earlier graphical calculus exists for Gaussian pure states [81].Circuits can be decomposed into a series of primitives such as quantum lattice gates [82].

Notes

For an introduction to continuous-variable quantum systems, see reviews [2,83–88] and books [18,89,90].See video tutorial by V. V. Albert.Bosonic states are typically represented with the assumption that a common phase reference exists, and the superselection rule compliant (SSRC) framework yields expressions without this assumption [91–96].

Cousins

  • Analog code— Bosonic codes are quantum version of analog codes.
  • \(t\)-design— Gaussian states, under a particular measure, do not form rigged two-designs [97].
  • Bosonic c-q code— Bosonic c-q codes are bosonic codes designed to transmit classical information.
  • EA bosonic code— EA bosonic codes utilize additional ancillary modes in a pre-shared entangled state, but reduce to ordinary bosonic codes when said modes are interpreted as noiseless physical modes.
  • Fermion code— Bosonic (fermionic) codes are associated with bosonic (fermionic) degrees of freedom.

Member of code lists

Primary Hierarchy

Quantum Domain
Bosonic Kingdom
Parents
Group-based quantum codeQECC Quantum
Group quantum codes whose physical spaces are constructed using the group of the reals \(\mathbb{R}\) under addition are bosonic codes.
Bosonic code
Children
Coherent-state constellation codeQuantum lattice
Numerically optimized bosonic code
Hybrid qudit-oscillator code
The physical Hilbert space of a hybrid qubit-oscillator code contains at least one oscillator.
Concatenated bosonic code
Oscillator-into-oscillator codeAnalog stabilizer
Oscillator-into-oscillator codes are bosonic codes with an infinite-dimensional logical subspace.
Qudit-into-oscillator codeFock-state bosonic
Qudit-into-oscillator codes are bosonic codes with a finite-dimensional logical subspace.
Bosonic stabilizer codeAnalog stabilizer Quantum lattice
Tiger code
Tiger codewords are superpositions of coherent states with the same energy, but coherent states are not eigenstates of the energy Hamiltonian. The AD Kraus operator \(E_{0}^{\otimes n}\) acts identically on each coherent state by shrinking the radius of the QSC's sphere.
Homological number-phase code
Homological number-phase codes are bosonic codes encoding logical qudits and/or logical rotors.
Penrose tiling code
Penrose tiling codes encode information into Penrose tilings, which are non-periodic tilings of \(\mathbb{R}^n\).
Qutrit-Pauli group-representation code

References

[1]
L. E. Ballentine, Quantum Mechanics (WORLD SCIENTIFIC, 2013) DOI
[2]
V. V. Albert, “Bosonic coding: introduction and use cases”, (2022) arXiv:2211.05714
[3]
A. M. Perelomov, “On the completeness of a system of coherent states”, (2002) arXiv:math-ph/0210005
[4]
V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, “On the completeness of the coherent states”, Reports on Mathematical Physics 2, 221 (1971) DOI
[5]
H. Bacry, A. Grossmann, and J. Zak, “Proof of completeness of lattice states in thekqrepresentation”, Physical Review B 12, 1118 (1975) DOI
[6]
M. Boon and J. Zak, “Discrete coherent states on the von Neumann lattice”, Physical Review B 18, 6744 (1978) DOI
[7]
E. Wigner, “On the Quantum Correction For Thermodynamic Equilibrium”, Physical Review 40, 749 (1932) DOI
[8]
A. Royer, “Wigner function as the expectation value of a parity operator”, Physical Review A 15, 449 (1977) DOI
[9]
J. Davis, N. Fabre, and U. Chabaud, “Identifying quantum resources in encoded computations”, (2024) arXiv:2407.18394
[10]
R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group”, Physical Review A 37, 3028 (1988) DOI
[11]
S. L. Braunstein and P. van Loock, “Quantum information with continuous variables”, Reviews of Modern Physics 77, 513 (2005) arXiv:quant-ph/0410100 DOI
[12]
A. Ferraro, S. Olivares, and M. G. A. Paris, “Gaussian states in continuous variable quantum information”, (2005) arXiv:quant-ph/0503237
[13]
A. Serafini, O. C. O. Dahlsten, D. Gross, and M. B. Plenio, “Canonical and micro-canonical typical entanglement of continuous variable systems”, Journal of Physics A: Mathematical and Theoretical 40, 9551 (2007) arXiv:quant-ph/0701051 DOI
[14]
X. WANG, T. HIROSHIMA, A. TOMITA, and M. HAYASHI, “Quantum information with Gaussian states”, Physics Reports 448, 1 (2007) arXiv:0801.4604 DOI
[15]
C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information”, Reviews of Modern Physics 84, 621 (2012) arXiv:1110.3234 DOI
[16]
L. Hackl and E. Bianchi, “Bosonic and fermionic Gaussian states from Kähler structures”, SciPost Physics Core 4, (2021) arXiv:2010.15518 DOI
[17]
T. Guaita, L. Hackl, and T. Quella, “Representation theory of Gaussian unitary transformations for bosonic and fermionic systems”, (2024) arXiv:2409.11628
[18]
A. Serafini, “Quantum Continuous Variables”, (2017) DOI
[19]
K. E. Cahill and R. J. Glauber, “Ordered Expansions in Boson Amplitude Operators”, Physical Review 177, 1857 (1969) DOI
[20]
M. de Gosson, Symplectic Geometry and Quantum Mechanics (Birkhäuser Basel, 2006) DOI
[21]
A. G. Burchards, “Continuous-Variable Quantum MacWilliams Identities”, (2025) arXiv:2502.09514
[22]
M. Y. Niu, I. L. Chuang, and J. H. Shapiro, “Hardware-efficient bosonic quantum error-correcting codes based on symmetry operators”, Physical Review A 97, (2018) arXiv:1709.05302 DOI
[23]
L. Susskind and J. Glogower, “Quantum mechanical phase and time operator”, Physics Physique Fizika 1, 49 (1964) DOI
[24]
J. Bergou and B.-G. Englert, “Operators of the phase. Fundamentals”, Annals of Physics 209, 479 (1991) DOI
[25]
S. D. Bartlett, H. de Guise, and B. C. Sanders, “Quantum encodings in spin systems and harmonic oscillators”, Physical Review A 65, (2002) arXiv:quant-ph/0109066 DOI
[26]
Stein, E. M. "Real variable methods, orthogonality, and oscillatory integrals." Princeton Math. Series 43 (1993).
[27]
M. M. Wolf, D. Pérez-García, and G. Giedke, “Quantum Capacities of Bosonic Channels”, Physical Review Letters 98, (2007) arXiv:quant-ph/0606132 DOI
[28]
L. Lami and M. M. Wilde, “Exact solution for the quantum and private capacities of bosonic dephasing channels”, Nature Photonics 17, 525 (2023) arXiv:2205.05736 DOI
[29]
J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001) arXiv:quant-ph/0105058 DOI
[30]
K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
[31]
S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications”, Nature Communications 8, (2017) arXiv:1510.08863 DOI
[32]
B. Demoen, P. Vanheuverzwijn, and A. Verbeure, “Completely positive quasi-free maps of the CCR-algebra”, Reports on Mathematical Physics 15, 27 (1979) DOI
[33]
J. Eisert and M. M. Wolf, “Gaussian quantum channels”, (2005) arXiv:quant-ph/0505151
[34]
M. M. Wolf, “Not-So-Normal Mode Decomposition”, Physical Review Letters 100, (2008) arXiv:0707.0604 DOI
[35]
F. Caruso, J. Eisert, V. Giovannetti, and A. S. Holevo, “Multi-mode bosonic Gaussian channels”, New Journal of Physics 10, 083030 (2008) arXiv:0804.0511 DOI
[36]
S. Pirandola, R. García-Patrón, S. L. Braunstein, and S. Lloyd, “Direct and Reverse Secret-Key Capacities of a Quantum Channel”, Physical Review Letters 102, (2009) arXiv:0809.3273 DOI
[37]
A. S. Holevo, “The Choi–Jamiolkowski forms of quantum Gaussian channels”, Journal of Mathematical Physics 52, (2011) arXiv:1004.0196 DOI
[38]
F. Caruso, J. Eisert, V. Giovannetti, and A. S. Holevo, “Optimal unitary dilation for bosonic Gaussian channels”, Physical Review A 84, (2011) arXiv:1009.1108 DOI
[39]
J. S. Ivan, K. K. Sabapathy, and R. Simon, “Operator-sum representation for bosonic Gaussian channels”, Physical Review A 84, (2011) arXiv:1012.4266 DOI
[40]
A. S. Holevo and R. F. Werner, “Evaluating capacities of Bosonic Gaussian channels”, (1999) arXiv:quant-ph/9912067
[41]
M. Takeoka, S. Guha, and M. M. Wilde, “The Squashed Entanglement of a Quantum Channel”, IEEE Transactions on Information Theory 60, 4987 (2014) arXiv:1310.0129 DOI
[42]
M. Takeoka, S. Guha, and M. M. Wilde, “Fundamental rate-loss tradeoff for optical quantum key distribution”, Nature Communications 5, (2014) arXiv:1504.06390 DOI
[43]
K. Goodenough, D. Elkouss, and S. Wehner, “Assessing the performance of quantum repeaters for all phase-insensitive Gaussian bosonic channels”, New Journal of Physics 18, 063005 (2016) arXiv:1511.08710 DOI
[44]
M. M. Wilde, M. Tomamichel, and M. Berta, “Converse Bounds for Private Communication Over Quantum Channels”, IEEE Transactions on Information Theory 63, 1792 (2017) arXiv:1602.08898 DOI
[45]
C. Ottaviani, R. Laurenza, T. P. W. Cope, G. Spedalieri, S. L. Braunstein, and S. Pirandola, “Secret key capacity of the thermal-loss channel: improving the lower bound”, SPIE Proceedings 9996, 999609 (2016) arXiv:1609.02169 DOI
[46]
S. Pirandola, S. L. Braunstein, R. Laurenza, C. Ottaviani, T. P. W. Cope, G. Spedalieri, and L. Banchi, “Theory of channel simulation and bounds for private communication”, Quantum Science and Technology 3, 035009 (2018) arXiv:1711.09909 DOI
[47]
N. Davis, M. E. Shirokov, and M. M. Wilde, “Energy-constrained two-way assisted private and quantum capacities of quantum channels”, Physical Review A 97, (2018) arXiv:1801.08102 DOI
[48]
G. Wang, C. Ottaviani, H. Guo, and S. Pirandola, “Improving the lower bound to the secret-key capacity of the thermal amplifier channel”, The European Physical Journal D 73, (2019) arXiv:1807.05402 DOI
[49]
F. A. Mele, L. Lami, and V. Giovannetti, “Maximum tolerable excess noise in continuous-variable quantum key distribution and improved lower bound on two-way capacities”, Nature Photonics (2025) arXiv:2303.12867 DOI
[50]
F. A. Mele, G. D. Palma, M. Fanizza, V. Giovannetti, and L. Lami, “Optical Fibers With Memory Effects and Their Quantum Communication Capacities”, IEEE Transactions on Information Theory 70, 8844 (2024) arXiv:2309.17066 DOI
[51]
K. Noh, S. M. Girvin, and L. Jiang, “Encoding an Oscillator into Many Oscillators”, Physical Review Letters 125, (2020) arXiv:1903.12615 DOI
[52]
J. Wu, A. J. Brady, and Q. Zhuang, “Optimal encoding of oscillators into more oscillators”, Quantum 7, 1082 (2023) arXiv:2212.11970 DOI
[53]
F. A. Mele, G. Barbarino, V. Giovannetti, and M. Fanizza, “Achievable rates in non-asymptotic bosonic quantum communication”, (2025) arXiv:2502.05524
[54]
F. A. Mele, G. Barbarino, V. Giovannetti, and M. Fanizza, “Non-asymptotic quantum communication on lossy transmission lines with memory”, (2025) arXiv:2503.13207
[55]
M. Moshinsky and C. Quesne, “Linear Canonical Transformations and Their Unitary Representations”, Journal of Mathematical Physics 12, 1772 (1971) DOI
[56]
Wagner, M. Unitary transformations in solid state physics. Netherlands.
[57]
S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient Classical Simulation of Continuous Variable Quantum Information Processes”, Physical Review Letters 88, (2002) arXiv:quant-ph/0109047 DOI
[58]
V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, “Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation”, New Journal of Physics 15, 013037 (2013) arXiv:1210.1783 DOI
[59]
A. Mari and J. Eisert, “Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient”, Physical Review Letters 109, (2012) arXiv:1208.3660 DOI
[60]
Y. Lim and C. Oh, “Efficient classical algorithms for linear optical circuits”, (2025) arXiv:2502.12882
[61]
U. Chabaud, G. Ferrini, F. Grosshans, and D. Markham, “Classical simulation of Gaussian quantum circuits with non-Gaussian input states”, Physical Review Research 3, (2021) arXiv:2010.14363 DOI
[62]
B. Dias and R. König, “Classical simulation of non-Gaussian bosonic circuits”, Physical Review A 110, (2024) arXiv:2403.19059 DOI
[63]
A. Barthe, M. Cerezo, A. T. Sornborger, M. Larocca, and D. García-Martín, “Gate-Based Quantum Simulation of Gaussian Bosonic Circuits on Exponentially Many Modes”, Physical Review Letters 134, (2025) arXiv:2407.06290 DOI
[64]
S. Lloyd and S. L. Braunstein, “Quantum Computation over Continuous Variables”, Physical Review Letters 82, 1784 (1999) arXiv:quant-ph/9810082 DOI
[65]
F. Arzani, R. I. Booth, and U. Chabaud, “Can effective descriptions of bosonic systems be considered complete?”, (2025) arXiv:2501.13857
[66]
D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001) arXiv:quant-ph/0008040 DOI
[67]
F. M. Miatto and N. Quesada, “Fast optimization of parametrized quantum optical circuits”, Quantum 4, 366 (2020) arXiv:2004.11002 DOI
[68]
T. Kalajdzievski, C. Weedbrook, and P. Rebentrost, “Continuous-variable gate decomposition for the Bose-Hubbard model”, Physical Review A 97, (2018) arXiv:1801.06565 DOI
[69]
T. Kalajdzievski and J. M. Arrazola, “Exact gate decompositions for photonic quantum computing”, Physical Review A 99, (2019) arXiv:1811.10651 DOI
[70]
R. A. Fisher, M. M. Nieto, and V. D. Sandberg, “Impossibility of naively generalizing squeezed coherent states”, Physical Review D 29, 1107 (1984) DOI
[71]
S. L. Braunstein and R. I. McLachlan, “Generalized squeezing”, Physical Review A 35, 1659 (1987) DOI
[72]
Y. Tong, V. V. Albert, J. R. McClean, J. Preskill, and Y. Su, “Provably accurate simulation of gauge theories and bosonic systems”, Quantum 6, 816 (2022) arXiv:2110.06942 DOI
[73]
U. Chabaud, M. Joseph, S. Mehraban, and A. Motamedi, “Bosonic Quantum Computational Complexity”, (2024) arXiv:2410.04274
[74]
U. Chabaud, D. Markham, and F. Grosshans, “Stellar Representation of Non-Gaussian Quantum States”, Physical Review Letters 124, (2020) arXiv:1907.11009 DOI
[75]
A. B. Michelsen, F. K. Marqversen, and M. Kastoryano, “Functional matrix product state simulation of continuous variable quantum circuits”, (2025) arXiv:2504.05860
[76]
R.-B. Wu, T.-J. Tarn, and C.-W. Li, “Smooth controllability of infinite-dimensional quantum-mechanical systems”, Physical Review A 73, (2006) arXiv:quant-ph/0505063 DOI
[77]
A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography”, Reviews of Modern Physics 81, 299 (2009) arXiv:quant-ph/0511044 DOI
[78]
Y. Xu, Y. Wang, and V. V. Albert, “Multimode rotation-symmetric bosonic codes from homological rotor codes”, Physical Review A 110, (2024) arXiv:2311.07679 DOI
[79]
H. Nagayoshi, W. Asavanant, R. Ide, K. Fukui, A. Sakaguchi, J. Yoshikawa, N. C. Menicucci, and A. Furusawa, “ZX Graphical Calculus for Continuous-Variable Quantum Processes”, (2024) arXiv:2405.07246
[80]
R. A. Shaikh, L. Yeh, and S. Gogioso, “The Focked-up ZX Calculus: Picturing Continuous-Variable Quantum Computation”, (2024) arXiv:2406.02905
[81]
N. C. Menicucci, S. T. Flammia, and P. van Loock, “Graphical calculus for Gaussian pure states”, Physical Review A 83, (2011) arXiv:1007.0725 DOI
[82]
L. Guo, T. Huang, and L. Du, “Engineering Bosonic Codes with Quantum Lattice Gates”, (2025) arXiv:2410.17069
[83]
C. Fabre and N. Treps, “Modes and states in quantum optics”, Reviews of Modern Physics 92, (2020) arXiv:1912.09321 DOI
[84]
B. M. Terhal, J. Conrad, and C. Vuillot, “Towards scalable bosonic quantum error correction”, Quantum Science and Technology 5, 043001 (2020) arXiv:2002.11008 DOI
[85]
A. Joshi, K. Noh, and Y. Y. Gao, “Quantum information processing with bosonic qubits in circuit QED”, Quantum Science and Technology 6, 033001 (2021) arXiv:2008.13471 DOI
[86]
W. Cai, Y. Ma, W. Wang, C.-L. Zou, and L. Sun, “Bosonic quantum error correction codes in superconducting quantum circuits”, Fundamental Research 1, 50 (2021) arXiv:2010.08699 DOI
[87]
K. Noh, “Quantum Computation and Communication in Bosonic Systems”, (2021) arXiv:2103.09445
[88]
S. M. Girvin, “Introduction to quantum error correction and fault tolerance”, SciPost Physics Lecture Notes (2023) arXiv:2111.08894 DOI
[89]
N. J. Cerf, G. Leuchs, and E. S. Polzik, “Quantum Information with Continuous Variables of Atoms and Light”, (2007) DOI
[90]
A. F. Kockum, A. Soro, L. García-Álvarez, P. Vikstål, T. Douce, G. Johansson, and G. Ferrini, “Lecture notes on quantum computing”, (2025) arXiv:2311.08445
[91]
Y. Aharonov and L. Susskind, “Charge Superselection Rule”, Physical Review 155, 1428 (1967) DOI
[92]
K. Mølmer, “Optical coherence: A convenient fiction”, Physical Review A 55, 3195 (1997) DOI
[93]
B. C. Sanders, S. D. Bartlett, T. Rudolph, and P. L. Knight, “Photon-number superselection and the entangled coherent-state representation”, Physical Review A 68, (2003) arXiv:quant-ph/0306076 DOI
[94]
S. D. BARTLETT, T. RUDOLPH, and R. W. SPEKKENS, “DIALOGUE CONCERNING TWO VIEWS ON QUANTUM COHERENCE: FACTIST AND FICTIONIST”, International Journal of Quantum Information 04, 17 (2006) arXiv:quant-ph/0507214 DOI
[95]
B. C. Sanders, “Review of entangled coherent states”, Journal of Physics A: Mathematical and Theoretical 45, 244002 (2012) arXiv:1112.1778 DOI
[96]
E. Descamps, A. Saharyan, A. Chivet, A. Keller, and P. Milman, “Modes, states and superselection rules in quantum optics and quantum information”, (2025) arXiv:2501.03943
[97]
R. Blume-Kohout and P. S. Turner, “The Curious Nonexistence of Gaussian 2-Designs”, Communications in Mathematical Physics 326, 755 (2014) arXiv:1110.1042 DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: oscillators

Cite as:
“Bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/oscillators
BibTeX:
@incollection{eczoo_oscillators, title={Bosonic code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/oscillators} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/oscillators

Cite as:

“Bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/oscillators

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/oscillators.yml.